CHEBYSHEV SPECTRAL-FINITE ELEMENTMETHOD FOR THREE-DIMENSIONALVORTICITY EQUATION

1.TheSchemesInthispaper,weconsidercombinedChebyshevspectraLfiniteelementmethodforthreedimensionalunsteadyvorticityequation.LetQbeaconvexpolygoninRZandIbetheinterval(--1,1).x~(xl,x2)andfi={(x,y)/xEQ,yEI}.Theboundaryoffiisdenotedbyoff.Denotethevorticityvectorandstreamvectorby(andoprespectively.Theircomponentsaref(q)andop(q),q=1,2,3.Letu>0bethekineticviscosity.fi,fZandfoaregivenvectors.Thethree-dimensionalvorticityequationisAssumethattheboundaryisafixednon-slipwallandsoop=oonafl.FOrsimpli…     

FOURIER－CHEBYSHEV SPECTRAL METHOD FOR SOLVING THREE－DIMENSIONAL VORTICITY EQUATION

ＦＯＵＲＩＥＲ－ＣＨＥＢＹＳＨＥＶＳＰＥＣＴＲＡＬＭＥＴＨＯＤＦＯＲＳＯＬＶＩＮＧＴＨＲＥＥ－ＤＩＭＥＮＳＩＯＮＡＬＶＯＲＴＩＣＩＴＹＥＱＵＡＴＩＯＮＧＵＯＢＥＮＹＵ（郭本瑜）；ＬＩＪＩＡＮ（李健）；ＭＡＨＥＰＩＮＧ（马和平）（Ｄｅｐａｒｔｍｅｎｔｏ...     

On blowups of vorticity for the homogeneous Euler equation

Blowups of vorticity for the three- and two-dimensional homogeneous Euler equations are studied. Two regimes of approaching a blow-up point, respectively, with variable or fixed time are analyzed. It is shown that in the n-dimensional ( $n=2,3$) generic case the blowups of degrees $1,\cdots ,n$ at the variable time regime and of degrees $1/2,\cdots ,(n+1)/(n+2)$ at the fixed time regime may exist. Particular situations when the vorticity blows while the direction of the vorticity vector is concentrated in one or two directions are realizable.     

On some cycles in linearized Wenger graphs

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements for prime power $q$ and let $p$ be the character of ${\mathbb{F}}_q$. For any positive integer $m$, the linearized Wenger graph $L_m\left(q\right)$ is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the $(m+1)$-dimensional vector space ${\mathbb{F}}_q^{m+1}$, and two vertices $p=\left(p\left(1\right),\dots ,p(m+1)\right)$ and $l=\left[l\left(1\right),\dots ,l(m+1)\right]$ being adjacent if $p\left(i\right)+l\left(i\right)=p\left(1\right){\left(l\left(1\right)\right)}^{p^{i?2}}$, for all $i=2,3,\dots ,m+1$. In this paper, we show that for any positive integers $m$ and $k$ with $3\le k\le p^2$, $L_m\left(q\right)$ contains even cycles of length $2k$ which is an open problem put forward by Cao et al. (2015).     

On certain linearized polynomials with high degree and kernel of small dimension

Let f be the ${\mathbb{F}}_q$-linear map over ${\mathbb{F}}_{q^{2n}}$ defined by $x?x+a{x}^{q^s}+b{x}^{q^{n+s}}$ with $\gcd ?(n,s)=1$. It is known that the kernel of f has dimension at most 2, as proved by Csajbók et al. in [9]. For n big enough, e.g. $n\ge 5$ when $s=1$, we classify the values of $b/a$ such that the kernel of f has dimension at most 1. To this aim, we translate the problem into the study of some algebraic curves of small degree with respect to the degree of f; this allows to use intersection theory and function field theory together with the Hasse-Weil bound. Our result implies a non-scatteredness result for certain high degree scattered binomials, and the asymptotic classification of a family of rank metric codes.     

Curvature computations for a two-component Camassa-Holm equation with vorticity

In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the computation of the sectional curvature K of the underlying configuration manifold. As a further result, we show that there are directions for which K is strictly positive and bounded away from zero.     

Solitary wave solutions of the modified equal width wave equation

In this paper we use a linearized numerical scheme based on finite difference method to obtain solitary wave solutions of the one-dimensional modified equal width (MEW) equation. Two test problems including the motion of a single solitary wave and the interaction of two solitary waves are solved to demonstrate the efficiency of the proposed numerical scheme. The obtained results show that the proposed scheme is an accurate and efficient numerical technique in the case of small space and time steps. A stability analysis of the scheme is also investigated.     

Well-posedness of the linearized Prandtl equation around a non-monotonic shear flow

In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class $2?\theta$ for any $\theta >0$. This result is almost optimal by the ill-posedness result proved by Gérard-Varet and Dormy, who construct a class of solution with the growth like $e^{\sqrt{k}t}$ for the linearized Prandtl equation around a non-monotonic shear flow.     

On the uniqueness of the solution of the two‐dimensional Navier–Stokes equation with a Dirac mass as initial vorticity

We propose two different proofs of the fact that Oseen's vortex is the unique solution of the two‐dimensional Navier–Stokes equation with a Dirac mass as initial vorticity. The first argument, due to C. E. Wayne and the second named author, is based on an entropy estimate for the vorticity equation in self‐similar variables. The second proof is new and relies on symmetrization techniques for parabolic equations. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chebyshev‐Legendre spectral method for solving the two‐dimensional vorticity equations with homogeneous Dirichlet conditions

The Chebyshev‐Legendre spectral method for the two‐dimensional vorticity equations is considered. The Legendre Galerkin Chebyshev collocation method is used with the Chebyshev‐Gauss collocation points. The numerical analysis results under the L²‐norm for the Chebyshev‐Legendre method of one‐dimensional case are generalized into that of the two‐dimensional case. The stability and optimal order convergence of the method are proved. Numerical results are given. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the nonlinear stability and the existence of selective decay states of 3D quasi-geostrophic potential vorticity equation

In this article, we study the dynamics of large-scale motion in atmosphere and ocean governed by the 3D quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions that are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D QGPV equation. The selective decay states are the 3D Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D QGPV system and in the case of 3D QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for nonequilibrium reversible-irreversible coupling.     

On the symmetry of equatorial travelling water waves with constant vorticity and stagnation points

The aim of this paper is to prove that equatorial travelling water waves at the surface of water flows with constant vorticity are symmetric, provided they have a profile that is monotonic between crests and troughs and that there are no stagnation points in the subsurface region.     

Harnack inequality for the linearized parabolic Monge-Ampère equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation

on parabolic sections associated with , under the assumption that the Monge-Ampère measure generated by satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group , where denotes the group of all invertible affine transformations on .

     

The temperature-jump problem based on the CES model of the linearized Boltzmann equation

An analytical discrete-ordinates method is used to solve the temperature-jump problem as defined by a synthetic-kernel model of the linearized Boltzmann equation. In particular, the temperature and density perturbations and the temperature-jump coefficient defined by the CES model equation are obtained (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented for general values of the accommodation coefficient to yield numerical results that compare well with solutions derived from more computationally intensive techniques.     

关于偶数阶中立型微分方程的线性化振动的一个注记

就偶数阶中立型时滞微分方程的所有有界解的振动性而言,研究了偶数阶非 线性中立型时滞微分方程［ｘ（ｔ）－ｐ（ｔ）ｘ（ｔ－τ）］－ｑ（ｔ）ｈ（ｘ（ｔ－σ））＝０和与它相应的 线性方程［ｘ（ｔ）－ｐ_０ｘ（ｔ－τ）］－ｑ-０ｘ（ｔ－τ）＝０在振动性上的等价性．所得结果改进 了 Ｌａｄａｓ Ｇ,和 Ｑｉａｎ Ｃ．的一个结果．     

Hamiltonians associated with the sixth Painlevé equation

We obtain the formula determining the general form of polynomial Hamiltonians associated with the sixth Painlevé equation and prove its uniqueness. We prove the existence of nonpolynomial Hamiltonians associated with this equation. We identify the Hamiltonian class for which the defining differential equation coincides with the equation (h-equation) for the simplest polynomial Hamiltonian (the Okamoto Hamiltonian). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 54–65, April, 2007.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.